In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

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1. Identify the matrix equation: \( \begin{bmatrix} 4 & -7 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -3 \\ 1 \end{bmatrix} \).
2. Recognize that this represents a system of linear equations.
3. The first row of the matrix equation corresponds to the equation: \( 4x - 7y = -3 \).
4. The second row of the matrix equation corresponds to the equation: \( 2x - 3y = 1 \).
5. Write the system of equations: \( \begin{cases} 4x - 7y = -3 \\ 2x - 3y = 1 \end{cases} \).

Key Concepts

Matrix Equation

A matrix equation is a mathematical expression that represents a system of linear equations in a compact form. It typically involves matrices and vectors, where the left side consists of a coefficient matrix multiplied by a variable vector, and the right side is a constant vector. Understanding how to interpret and manipulate these matrices is crucial for converting them into a system of equations.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed. They take the form ax + by = c, where a, b, and c are constants, and x and y are variables. In the context of a matrix equation, each row of the coefficient matrix corresponds to a linear equation, which can be extracted to form a system of equations.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. When converting a matrix equation into a system of equations, each equation represents a relationship between the variables, allowing for methods such as substitution or elimination to find solutions.